SOLUTION: The fourth term of a geometric progression exceeds the third term by 54, and the sum of the second and third term is 36. Find the common ratio if it is positive

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Question 988607: The fourth term of a geometric progression exceeds the third term by 54, and the sum of the second and third term is 36. Find the common ratio if it is positive
Found 3 solutions by MathLover1, KMST, MathTherapy:
Answer by MathLover1(20850)   (Show Source):
You can put this solution on YOUR website!
if the fourth term exceeds the third term by , we have
....eq.1
if the sum of the second and third term is , we have
....eq.2
By using we have:

....eq.1
....eq.2
---------------------------------------------------add eq.2 and eq.1

.............substitute in eq.2

....eq.2

......divide by

=> one solution will be: =>
=> another solution will be: =>=>
now we can find first term:
if
....eq.2

if :
....eq.2
......multiply by

so, there are two solutions:
1. and
2. and

then the second term is:
if and :

third term is

fourth term

the terms of sequence are:=>

if ,

.
third term is

and fourth term is

the terms of sequence are:=>

now, check given data:
for =>
....eq.1

....eq.2

for =>
....eq.1

....eq.2

Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!
= the common ratio.
= the second term.
= the third term.
= the fourth term.
The fourth term of a geometric progression exceeds the third term by 54 translates as
<--><--> .
The sum of the second and third term is 36 translates as
<--><--> .
The ratio both equations tells us that
-->-->-->}--> .
The equation is a quadratic equation.
As is true for all quadratic equations, it can be solved by "completing the square, or by using the quadratic formula.
This particular quadratic equation can also be solved by factoring:
--->--->}--->---> .
If the common ratio is positive, it must be .

Answer by MathTherapy(10557)   (Show Source):
You can put this solution on YOUR website!
The fourth term of a geometric progression exceeds the third term by 54, and the sum of the second and third term is 36. Find the common ratio if it is positive

Since , then , , and
Since exceeds by 54, then we can say that:

---------- eq (i)
Since the sum of and is 36, then we can say that:

--------- eq (ii)
Since and , we can then say that:
------ Cross-multiplying
-------- Factoring out GCF, 18

(r - 3)(2r + 1) = 0
Common ratio, or OR (ignore, since r MUST be > 0)